# Full text of "Treatise On Analysis Vol-Ii"

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```374       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

For all x, y in A, the mapping V\-*(V* ng(x)\ng(y)) is a continuous
linear form on j/'g and so, by virtue of the isomorphism of ^C(S^) and j#'g,
there exists (13.1) a (complex) measure \jLXy}> on S^ such that

(15.9.2.4)       g(zx, y) - (Ug(z) * ng(z) \ ng(y)) = f  /(z) rf/1,,,00

Js'0

for all 2- e A.

When OeS0, we have seen (15.9.2.2) that the function z : /H->/(Z) is
zero at the point 0; consequently, if mXty is the measure induced on Sg by
HXty (13.1.8), we can write in all cases

(15.9.2.5)                        g(zx, y)

It is clear that the measures mXt y are bounded. Moreover, for given jc and y
in A, the measure mXty is the only bounded measure ra' on Sg such that
g(zx> y) = J z(%) dm'(%) for all z 6 A, because the functions z are dense
in*g(S,)(1/.20.6).

If the formula (15.9.2.1) is true, then for all x, y, z in A we must have
g(zx, y) = f   2(x)x(x)\$(x) dmg(i). Comparing this with (15.9.2.5), what we

JSy

shall in fact prove is the existence of a measure mg on S^ such that

(15.9.2.6)                                mXiy=x\$'mg
for all jc, y in A (cf. (13.1.5)).

If we have constructed such a measure, then for all functions F e
we shall have

(15.9.2.7)                   F • mx>y = (fltf ) • mg = (tf) - mF ,

where we have put mF = F • IH,, which is a bounded measure (13.14.4). We
shall begin by constructing a linear mapping Fh-»wF of ^Tc(Sff) into M£(S0),
satisfying the equality

(15.9.2.8)                     F-m^.^-mp

for all x, y in A, and such that mF ^ 0 whenever F S| 0. Having achieved this,
the measure mg will be simply the linear form FH*WIF(!) (13.3.1).n each case, Sg is separable,
```